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\def\ohm{$\Omega \;$}
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\centerline{\bf Homework 06 $\;\;\;\;\;\;\;$ (Phys333 -- Fall 2016 -- R.Sonnenfeld)}

\subsubsection*{Due November 9, 2016 in class}
\subsection*{Instructions:}
Each problem should begin at the top of a new sheet of paper kept
in a binder.
Each problem should have your name on the left and the \emph{SPN} on
the right, circled.  
For numerical
answers, show numbers plugged into the equation before solving with a 
calculator.  Numerical answers should include SI units.
One star problems are easier than two star.  
%If you
%think a one star problem is very difficult you are
%probably missing a conceptual ``trick''.
\bigskip


\noindent {\bf * SPN 6--07.} Does it make economic sense to pay \$5 for a
compact fluorescent bulb when an incandescent only costs \$0.50?
Assume electricity costs \$0.12 per kilowatt hour and compare a 60~Watt
incandescent to a 15~Watt fluorescent (they both produce about the same
amount of light).  If both lightbulbs last about 10,000 hours, compare
the amount you ended up spending (purchase + electricity cost) on the 
two bulbs.\\

\noindent {\bf * SPN 6--08.} A 2.2-m-long, wire is one mm in diameter
and its resistivity is variable.
\begin{equation}
    \rho(x)=(3.8 \times 10^{-6} \; [1+(\frac{x}{1.0 m})^2] \; \Omega \; m
\end{equation}
Where $x$ is measured from one end of the wire.  How much current
does the wire carry if hooked up to a twelve volt battery?
\\

\noindent {\bf * SPN 6--09.} An RC circuit has an eight millisecond time
constant.  How long after discharge begins are
\begin{description}
    \item[(a)]{ 
        The charge on the capacitor reduced to half its initial value?    }
    \item[(b)]{ 
        The energy stored in the capacitor is reduced to half its initial
        value?
    }
\end{description}

\noindent {\bf ** SPN 6--10.} A capacitor charging circuit with a 50 $\mu$F
capacitor has a 40~ms time constant.  The capacitor has zero charge
at t=0 and the initial current to the capacitor
is 65~mA.  What is the capacitor's voltage after 20~ms?\\

\noindent {\bf * SPN 6--11.} All the light bulbs in the circuit below
are identical and the battery is ideal.  Rank the light bulbs from
brightest to dimmest (indicate any that are equal in brightness to
eachother).\\

\begin{figure}[h!]
    \begin{center}
     \begin{circuitikz}
   \draw (0,-2)
   to[battery,v=$\varepsilon$] (0,2) % The voltage source
   to[short] (4,2)
   to[lamp,l=$P$] (4,0); % The resistor

% Please note, the ; matter, they are "ends of phrases"
% All points are ref'd to previous points until you get a ;
% I think the ; resets to the origin.
  \draw (4,0)
   to[short] (5.0,0)
   to[lamp,l=$R$] (5.0,-2) % The resistor
   to[short] (0,-2)
   to[short] (0,0);
  \draw (4,0)
   to[short] (3.0,0)
   to[lamp,l=$Q$] (3.0,-2);
  \draw (4,2)
   to[short] (6.5,2) 
   to[lamp,l=$S$] (6.5,0) 
   to[lamp,l=$T$] (6.5,-2) 
   to[short] (6.5,-2) 
   to[short] (0,-2);
    \end{circuitikz}
%\caption{Switch S is open for a long time, then closed at t=0.}
\end{center}
\end{figure}

\noindent {\bf * SPN 6--12.} The capacitors below are fully charged and the
switch is closed at $t=0$. At what time has the current in the
8 k\ohm resistor decayed to half the value it had $t=0$?\\

\begin{figure}[h!]
    \begin{center}
     \begin{circuitikz}
   \draw (0,-4)
   to[short] (0,-3) 
   to[short] (2,-3) 
   to[C, l=20 $\mu F$] (2,0)  
   to[short] (-2,0)
   to[C, l=60 $\mu F$] (-2,-1.5)  
   to[C, l=60 $\mu F$] (-2,-3)  
   to[short] (-2,-3) 
   to[short] (0,-3);
   \draw (0,0)
   to[short] (0,1) 
   to[closing switch,l=S] (4,1)
   to[R, l=8 k\ohm] (6,1)
   to[short] (7,1) 
   to[short] (7,0)
   to[short] (8,0)
   to[R, l=20 k\ohm] (8,-3)
   to[short] (7,-3)
   to[short] (7,-4)
   to[short] (0,-4);
   \draw (7,0)
   to[short] (6,0)
   to[R, l=30 k\ohm] (6,-3)
   to[short] (7,-3);
    \end{circuitikz}
%\caption{Switch S is open for a long time, then closed at t=0.}
\end{center}
\end{figure}

\noindent {\bf ** SPN 6--13.} What is the equivalent resistance of the
Wheatstone bridge circuit below?  What is the current through each
resistor?  (Make a 5x5 matrix $\boldsymbol{A}$ using KVL and KCL.  Use
Matlab or Octave to invert for $I$ using $I=\boldsymbol{A} \backslash b$.)\\

\begin{figure}[h!]
    \begin{center}
     \begin{circuitikz}
   \draw (-4,-5)
   to[battery,v=$12 \; V$] (-4,1) % 
   to[short] (0,1)
   to[short] (0,0);
   \draw (-4,-5)
   to[short] (0,-5)
   to[short] (0,-4)
   to[R, l=200 \ohm] (-2,-2)
   to[R, l=150 \ohm] (0,0)   
   to[R, l=75 \ohm] (2,-2)   
   to[R, l=300 \ohm] (0,-4);
   \draw (-2,-2)
   to[R, l=100 \ohm] (2,-2);
       \end{circuitikz}
%\caption{Switch S is open for a long time, then closed at t=0.}
\end{center}
\end{figure}

%It turns out that the circuit below is topologically equivalent to the
%Wheatstone bridge circuit.
\noindent {\bf ** SPN 6--14.} Make a table of the current through  (and voltage
across) each
resistor in the circuit below.  Indicate with arrows the direction of
positive
current flow for each resistor.  (Make a 3x3 matrix $\boldsymbol{A}$ using KVL and KCL.  Use
Matlab or Octave to invert for $I$ using $I=\boldsymbol{A} \backslash b$.)\\

\begin{figure}[h!]
    \begin{center}
     \begin{circuitikz}
   \draw (0,0)
   to[battery,v=$5 \; V$] (0,2) % 
   to[short] (0,3) 
   to[R, l=10 \ohm] (0,5)
   to[short] (3,5) 
   to[R, l=8 \ohm] (6,5)
%   to[R, l=40 k\ohm] (3,5)
   to[battery,v=$9 \; V$] (6,0) % 
   to[short] (3,0) 
   to[short] (0,0) ;
   \draw (3,0)
   to[R, l=6 \ohm] (3,5);
    \end{circuitikz}
%\caption{Switch S is open for a long time, then closed at t=0.}
\end{center}
\end{figure}

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